Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

The All Different and Global Cardinality Constraints on Set, Multiset and Tuple Variables

We describe how the propagator for the All-Differentconstraintcanbegeneralizedtoprune variableswhosedomainsarenot justsimplefinitedomains. We show,forexample, howitcanbeused to propagate set variables, multisetvariablesandvariables whichrepresenttuplesofvalues. Wealsodescribehowthepropagatorfor theglobal cardinalityconstraint(whichisageneralization of the All-Different constraint) can be generalized in a similar way. Experiments show that such propagators can be beneficial in practice, especially when the domains are large

Symmetry breaking revisited

Abstract. Symmetries in constraint satisfaction problems (CSPs) are one of the difficulties that practitioners have to deal with. We present in this paper a new method based on the symmetries of decisions taken from the root of the search tree. This method can be seen as an improvement of the nogood recording presented by Focacci and Milano[2] and Fahle, Schamberger and Sellmann[1]. We present a simple formalization of our method for which we prove correctness and completeness results. We also show that our method is theoretically more efficient as the number of dominance checks, the number of nogoods and the size of each nogood are smaller. This is confirmed by an experimental evaluation on the social golfer problem, a very difficult and highly symmetrical real world problem. We are able to break all symmetries for problems with more than 10 36 symmetries. We report both new results, and

Constraints over ontologies

This paper presents a new constraint domain, where variables can be assigned values that are organised in a tree hierarchy. The introduction of this new constraint domain is motivated by applications for the configuration of product and services, for instance, in the context of e-commerce. The paper proposes a small constraint language, based on comparisons and the use of monotonic functions. An approximation of domains as convex sets is detailed and propagation rules for achieving bound-consistency on the constraints are reviewed. This new constraint domain is general, and the author hopes that it will be a useful tool not only for solving configuration problems but also for promoting constraint technology in new application domains related to content classification or the semantic web. 1. Modelling Problems with hierarchical values The theory of Constraint Programming has been developed in full generality, independently of value domains: constraint programmers can state problems where the variables are unknowns who should take their values among (integer, real

Hybrid Set Domains to Strengthen Constraint Propagation and Reduce Symmetries

International audienceFinite set constraints represent a natural choice to model configuration design problems using set cardinality and disjointness, covering or partition constraints over (families of) set variables. Such constraints are available in most set-based constraint languages, often in the form of n-ary decomposable constraints. The corresponding filtering algorithms make use of local bound consistency techniques. In this paper we show that when the set cardinality constraints are handled together with the n-ary constraints, and set variable domains are specified by set intervals, efficient global filtering algorithms can be derived. We consider the particular case of the n-ary disjoint constraint disjoint([X 1 ,. .. , X n ], [c 1 ,. .. , c n ]) for a family of pairwise disjoint sets X i of fixed cardinality c i. We present a set of conditions and inference rules to infer Bounds Consistency (BC), together with an efficient global filtering algorithm. We also explain why this level of pruning cannot be achieved with a common FD formulation based on the alldiff constraint, enriched with lexicographic ordering constraints; but is actually equivalent to a dual FD representation based on the Generalized Cardinality Constraint